#8071517

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4879 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(2,6)(3,8)(10,14)(15,16)(17,18,23,25)(19,26,21,24)(20,22,27,32)(28,31,30,29);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,19,23,21)(18,24,25,26)(20,28,27,30)(22,29,32,31);(2,6)(3,8)(18,25)(19,21)(22,32)(28,30);(3,8)(4,7)(17,20)(18,22)(19,28)(21,30)(23,27)(24,29)(25,32)(26,31);(1,3,5,8)(2,7,6,4)(9,11,13,12)(10,15,14,16)(17,21,23,19)(18,26,25,24)(20,30,27,28)(22,31,32,29);(1,4,5,7)(2,8,6,3)(9,12,13,11)(10,16,14,15)(17,22)(18,27)(19,29)(20,25)(21,31)(23,32)(24,30)(26,28)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7978713);